The circle of reducible solutions \(\mathbb {T}\)  is still present after the perturbations we have added. If for none of the \(B\in \mathbb {T}\)  the Dirac operator \(D_B\)  on \(M_{\varphi }\)  has kernel (i.e. by Lemma REF  we have \(H^0(L)=0\) ), we can conclude in the same way as in the case of \(S^1\times S^2\)  [1]}. Otherwise, in order to proceed choose a Morse function
\(f:\mathbb {T}\rightarrow \mathbb {R}\) 
